Skip to main content
SpringerLink
Log in

Another class of warped product submanifolds of Kenmotsu manifolds

  • Original Paper
  • Published:
Revista de la Real Academia de Ciencias Exactas, Físicas y Naturales. Serie A. Matemáticas Aims and scope Submit manuscript

Abstract

In this paper, we study semi-slant submanifolds and warped product semi-slant submanifolds of Kenmotsu manifolds, which have not been considered in earlier studies. A characterization and a general sharp inequality for the lower bound of the squared norm of second fundamental form of such immersions are obtained. The necessary condition is obtained for the warped products to satisfy the equality case of the derived inequality. We also give non-trivial examples of such submanifolds.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Al-Solamy, F.R., Khan, V.A.: Warped product semi-slant submanifolds of a Sasakian manifold. Serdica Math. J. 34, 597–606 (2008)

    MathSciNet  MATH  Google Scholar 

  2. Al-Solamy, F.R., Khan, V.A., Uddin, S.: Geometry of warped product semi-slant submanifolds of nearly Kaehler manifolds. Results. Math. 71, 783 (2017). doi:10.1007/s00025-016-0581-4

    Article  MathSciNet  MATH  Google Scholar 

  3. Arslan, K., Ezentas, R., Mihai, I., Murathan, C.: Contact CR-warped product submanifolds in Kenmotsu space forms. J. Korean Math. Soc. 42, 1101–1110 (2005)

    Article  MathSciNet  Google Scholar 

  4. Atceken, M.: Warped product semi-slant submanifolds in Kenmotsu manifolds. Turk. J. Math. 36, 319–330 (2012)

    Google Scholar 

  5. Atceken, M.: Contact CR-warped product submanifolds in Kenmotsu space forms. Bull. Iran Math. Soc. 39, 415–429 (2013)

    MathSciNet  MATH  Google Scholar 

  6. Bishop, R.L., O’Neill, B.: Manifolds of negative curvature. Trans. Am. Math. Soc. 145, 1–49 (1969)

    Article  MathSciNet  Google Scholar 

  7. Blair, D.E.: Contact Manifolds in Riemannian Geometry. Lecture Notes in Mathematics, vol. 509. Springer, New York (1976)

    Book  Google Scholar 

  8. Cabrerizo, J.L., Carriazo, A., Fernandez, L.M., Fernandez, M.: Semi-slant submanifolds of a Sasakian manifold. Geom. Dedic. 78, 183–199 (1999)

    Article  MathSciNet  Google Scholar 

  9. Cabrerizo, J.L., Carriazo, A., Fernandez, L.M., Fernandez, M.: Slant submanifolds in Sasakian manifolds. Glasgow Math. J. 42, 125–138 (2000)

    Article  MathSciNet  Google Scholar 

  10. Chen, B.-Y.: Geometry of warped product CR-submanifolds in Kaehler manifolds. Monatsh. Math. 133, 177–195 (2001)

    Article  MathSciNet  Google Scholar 

  11. Chen, B.-Y.: Geometry of warped product CR-submanifolds in Kaehler manifolds II. Monatsh. Math. 134, 103–119 (2001)

    Article  MathSciNet  Google Scholar 

  12. Chen, B.-Y.: Pseudo-Riemannian Geometry, \(\delta \)-Invariants and Applications. World Scientific, Hackensack (2011)

    Book  Google Scholar 

  13. Chen, B.-Y.: Geometry of warped product submanifolds: a survey. J. Adv. Math. Stud. 6(2), 1–43 (2013)

    MathSciNet  MATH  Google Scholar 

  14. Chen, B.-Y.: Differential Geometry of Warped Product Manifolds and Submanifolds. World Scientific, Hackensack (2017)

    Book  Google Scholar 

  15. Chen, B.-Y., Uddin, S.: Warped product pointwise bi-slant submanifolds of Kaehler manifolds. Publ. Math. Debrecen 92(1), 1–16 (2018)

    MathSciNet  Google Scholar 

  16. Hiepko, S.: Eine inner kennzeichungder verzerrten produkte. Math. Ann. 241, 209–215 (1979)

    Article  MathSciNet  Google Scholar 

  17. Kenmotsu, K.: A class of almost contact Riemannian manifolds. Tohoku Math. J. 24, 93–103 (1972)

    Article  MathSciNet  Google Scholar 

  18. Khan, K.A., Khan, V.A., Uddin, S.: Warped product submanifolds of cosymplectic manifolds. Balkan J. Geom. Appl. 13, 55–65 (2008)

    MathSciNet  MATH  Google Scholar 

  19. Kobayashi, M.: Semi-invariant submanifolds of a certain class of almost contact manifolds. Tensor (N.S.) 43, 28–36 (1986)

    MathSciNet  MATH  Google Scholar 

  20. Murathan, C., Arslan, K., Ezentas, R., Mihai, I.: Warped product submanifolds in Kenmotsu space forms. Taiwan. J. Math. 10, 1431–1441 (2006)

    Article  MathSciNet  Google Scholar 

  21. Mustafa, A., De, A., Uddin, S.: Characterization of warped product submanifolds in Kenmotsu manifolds. Balkan J. Geom. Appl. 20, 86–97 (2015)

    MathSciNet  MATH  Google Scholar 

  22. Naghi, M.F., Uddin, S., Al-Solamy, F.R.: Warped product submanifolds of Kenmotsu manifolds with slant fiber. Filomat, 1–12 (2017)

  23. Naghi, M.F., Uddin, S., Al-Solamy, F.R.: Geometry of warped product pseudo-slant submanifolds of Kenmotsu manifolds. Quaest. Math., 1–15 (2017)

  24. Olteanu, A.: Contact CR-doubly warped product submanifolds in Kenmotsu space forms. J. Ineq. Pure Appl. Math. 10(4), 7 (2009). (Article 119)

    MathSciNet  MATH  Google Scholar 

  25. Sahin, B.: Non existence of warped product semi-slant submanifolds of Kaehler manifolds. Geom. Dedic. 117, 195–202 (2006)

    Article  Google Scholar 

  26. Sahin, B.: Warped product pointwise semi-slant submanifolds of Kaehler manifolds. Port. Math. 70, 252–268 (2013)

    Article  Google Scholar 

  27. Uddin, S., Khan, V.A., Khan, K.A.: Warped product submanifolds of a Kenmotsu manifold. Turk. J. Math. 36, 319–330 (2012)

    MathSciNet  MATH  Google Scholar 

  28. Uddin, S., Al-Solamy, F.R.: Warped product pseudo-slant submanifolds of cosymplectic manifolds, An. Ştiinţ. Univ. Al. I. Cuza Iaşi Mat (N.S.) Tome LXIII (2016), \(f_2\) vol. 3, 901–913

  29. Uddin, S., Alghanemi, A., Naghi, M.F., Al-Solamy, F.R.: Another class of warped product CR-submanifolds in Kenmotsu manifolds. J. Math. Comput. Sci. 17, 148–157 (2017)

    Article  Google Scholar 

  30. Uddin, S.: Geometry of warped product semi-slant submanifolds of Kenmotsu manifolds. Bull. Math. Sci. (2017). doi:10.1007/s13373-017-0106-9

    Article  MATH  Google Scholar 

  31. Uddin, S., Al-Solamy, F.R.: Warped product pseudo-slant immersions in Sasakian manifolds. Publ. Math. Debrecen 91(2), 1–14 (2017)

    MathSciNet  MATH  Google Scholar 

  32. Uddin, S., Chen, B.-Y., Al-Solamy, F.R.: Warped product bi-slant immersions in Kaehler manifolds. Mediterr. J. Math. 14, 95 (2017). doi:10.1007/s00009-017-0896-8

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, Faculty of Science, King Abdulaziz University, Jidda, 21589, Saudi Arabia

    Siraj Uddin, Monia Fouad Naghi & Falleh Rijaullah Al-Solamy

Authors
  1. Siraj Uddin
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Monia Fouad Naghi
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Falleh Rijaullah Al-Solamy
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Siraj Uddin.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Uddin, S., Naghi, M.F. & Al-Solamy, F.R. Another class of warped product submanifolds of Kenmotsu manifolds. RACSAM 112, 1141–1155 (2018). https://doi.org/10.1007/s13398-017-0415-6

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s13398-017-0415-6

Keywords

Mathematics Subject Classification

Search

Navigation